By M. Hazewinkel (auth.), M. Hazewinkel (eds.)
This ENCYCLOPAEDIA OF arithmetic goals to be a reference paintings for all components of mathe matics. it's a translation with updates and editorial reviews of the Soviet Mathematical Encyclopaedia released through 'Soviet Encyclopaedia Publishing condo' in 5 volumes in 1977-1985. The annotated translation involves ten volumes together with a distinct index quantity. There are 3 types of articles during this ENCYCLOPAEDIA. to begin with there are survey-type articles facing many of the major instructions in arithmetic (where a slightly positive subdivi sion has been used). the most requirement for those articles has been that they need to provide a fairly entire updated account of the present situation in those parts and they can be maximally obtainable. normally, those articles will be comprehensible to arithmetic scholars of their first specialization years, to graduates from different mathematical parts and, counting on the explicit topic, to experts in different domain names of technology, en gineers and lecturers of arithmetic. those articles deal with their fabric at a pretty basic point and objective to provide an concept of the type of difficulties, recommendations and ideas inquisitive about the realm in query. in addition they include historical past and motivation instead of targeted statements of exact theorems with targeted definitions and technical info on how one can perform proofs and buildings. the second one form of article, of medium size, comprises extra exact concrete difficulties, effects and techniques.
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Since 19(1 0" N) =k + 19 N, the decimal logarithms of numbers that differ by a multiple of Hi' have the same mantissa and differ only in the characteristics. This property is the basis for the construction of logarithm tables, which contain only the mantissa of the logarithms of integers. 71828 ... ; they are denoted by inN. 30258· . 43429· ... See also Logarithmic function. Material of the article of the same name in BSE-3. Editorial comments. Western writers of post-calculus mathematics almost always use log N instead of In N to denote the natural logarithm (base e) of N.
A L h lk' . Sme'm AMS 1980 Subject Classification: 20E05 LocALLY FREE SHEAF - A sheaf of modules that is locally isomorphic to the direct sum of several copies of the structure sheaf. More precisely, let (X, aJ x) be a ringed space. A sheaf of modules ~ over aJ x is said to be locally free if for every point x EX there is an open neighbourhood U C X, X E U, such that the restriction ~ Iu of ~ to U is a free sheaf of modules over aJ x Iu, that is, it is isomorphic to the direct sum of a set I(x) of copies of the structure sheaf aJ x Iu.
Baire theorem). , countable inductive limits of Frechet spaces), nuclear, Schwartz and Montel spaces. Topological tensor products were introduced as a tool to study spaces of operators and spaces of vector-valued functions and distributions. A. Grothendieck [A4] investigated the nuclear spaces in this context and raised the approximation problem which was solved by P. Enflo  by giving the first example of a Banach space without the approximation property. Later on, A. Szankowski proved that the space of all bounded linear operators on a Hilbert space does not have the approximation property.